Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

Abstract

We prove a strict monotonicity theorem for first passage percolation on any Cayley graph of a virtually nilpotent group which is not isomorphic to the standard Cayley graph of $\mathbb{Z}$: given two distributions ν and $\tilde{\mathit{\nu }}$ with finite mean, if $\tilde{\mathit{\nu }}$ is strictly more variable than ν and ν is subcritical in an appropriate sense, then the expected passage times associated to ν exceed those of $\tilde{\mathit{\nu }}$ by an amount proportional to the graph distance. This generalizes a theorem of van den Berg and Kesten from 1993, which treats the standard Cayley graphs of ${\mathbb{Z}}^{\mathit{d}},\mathit{d}\ge 2$. In fact, our theorem applies to any bounded-degree graph which either is of strict polynomial growth or is quasi-isometric to a tree and which satisfies a certain geometric condition we call “admitting detours.” If a bounded degree graph does not admit detours, such a strict monotonicity theorem with respect to variability cannot hold.

Moreover, we show that for the same class of graphs, independent of whether the graph admits detours, any appropriately subcritical weight measure is “absolutely continuous with respect to the expected empirical measure of the geodesic.” This implies a strict monotonicity theorem with respect to stochastic domination.